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Topic 1

Introduction toNonlinear Analysis

• Introduction to the course

• The importance of nonlinear analysis

• Four illustrative films depicting actual and potentialnonlinear analysis applications

• General recommendations for nonlinear analysis

• Modeling of problems

• Classification of nonlinear analyses

• Example analysis of a bracket, small and largedeformations, elasto-plastic response

• Two computer-plotted animations-elasto-plastic large deformation response of a platewith a hole-large displacement response of a diamond-shapedframe

• The basic approach of an incremental solution

• Time as a variable in static and dynamic solutions

• The basic incremental/iterative equations

• A demonstrative static and dynamic nonlinear analysisof a shell

Section 6.1

6.1,6.2,6.3,6.4

The shell analysis is reported in

Ishizaki, T., and K. J. Bathe, "On Finite Element Large Displacementand Elastic-Plastic Dynamic Analysis of Shell Structures," Computers& Structures, 12, 309-318, 1980.

FIELD OF NONLINEAR

ANAL'fS1S

• (ONTINLAVl.tJ'\ MECHANICS

• FIN liE Ell:: t-"\ENT D15

CR~.,..'"2A-TION S

• NLAt-'\ERIc.AL

AL5o"R 11\-\ MS

• SOfTWA'QE

(ONS' 't>E:'RAT\ DN~

Nt::- CONCENTRATE

ON ~-• M&Tl-\o1:»TI-\Po-T A'e&

5ENE'RA LL'I A?PLI c.AELE

• !'"10t>E'RtJ 'E"(\..HJ I QlA.E S

• "PRACTICAL ?ROCEblA~E:S

~IHE1Hc>'t>s IItA-\ Al'f OR

ARE NOw ~E(OMIN b AN

INTE G1</\ l 'flPt'RI 0 rCAl:> leAf SO~TWAKE

Topic One 1-3

'R~IEF OVERVI£~

Ol= CouRSE

• 6fOMElRIC A Nt>

,""A/ER.AL NONLlNE.AR

ANAL%IS

• S"1AT,c: AN~ :b'fAlAM1C

SOL IAllONS.

• EASIC r~ll'I/c..rt>LE:,,;)

ANb TIi~IR l.A<;E

WILL EE OF INIERES,T

IN MAN 'I E;K AN( ~ E So OF

ENSINl:.£~IN(; 11-\"i:ol..lb t\

0lAi TIi'E WO'RL\::>

Markerboard1-1

IN ,HIS LE:CTUR~

WE "bl'>[\)\><; $ol'\E

IN TRO"t:>l,\CTDR..., Vlf\V

G'l:.A'PItS AN!) S HoW

<; C' nE <; It 01':., t-1 0\1 \ES

WE TH-EN CLASS\f'l

NON.LINE~'R AI\}AL'-/'SES

WE ~1.sCW;C::; THE

\SAS IC A??'KOACI-\ I:JF

AN INCKEMENTA L

$OLlAi\ON

W( bl\lE EXAtWLES

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1-4 Introduction to Nonlinear Analysis

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FINITE ELEMENTNONLINEAR ANALVSIS

• Nonlinear analysis in engineeringmechanics can be an art.

• Nonlinear analysis can bea frustration.

• It always is a great challenge.

Some important engineeringphenomena can only be assessed onthe basis of a nonlinear analysis:

• Collapse or buckling of structuresdue to sudden overloads

• Progressive damage behavior due tolong lasting severe loads

• For certain structures (e.g. cables),nonlinear phenomena need beincluded in the analysis even forservice load calculations.

The need for nonlinear analysis hasincreased in recent years due to theneed for

- use of optimized structures

- use of new materials

- addressing safety-related issues ofstructures more rigorously

The corresponding benefits can bemost important.

Problems to be addressed by a nonlinear finite element analysis are foundin almost all branches of engineering,most notably in,

Nuclear EngineeringEarthquake EngineeringAutomobile IndustriesDefense IndustriesAeronautical EngineeringMining IndustriesOffshore Engineering

and so on

Topic One 1-5

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Film InsertArmoredFightingVehicleCourtesy of GeneralElectricCAE International Inc.

Topic One 1-7

Film InsertAutomobileCrashTestCourtesy ofFord OccupantProtection Systems


Film InsertEarthquakeAnalysisCourtesy ofASEA Researchand InnovationTransformersDivision

Topic One 1-9

Film InsertTacomaNarrowsBridgeCollapseCourtesy ofBarney D.Elliot


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For effective nonlinear analysis,a good physical and theoreticalunderstanding is most important.

PHYSICAL MATHEMATICALINSIGHT FORMULATION

4

(INTERACTION AND )

MUTUAL ENRICHMENT

BEST APPROACH

• Use reliable and generally applicablefinite elements.

• With such methods, we can establishmodels that we understand.

• Start with simple models (of nature)and refine these as need arises.

4

A "PHILOSOPHY" FOR PERFORMINGA NONLINEAR ANALYSIS

TO PERFORM A NONLINEARANALYSIS

• Stay with relatively small and reliable models.

• Perform a linear analysis first.

• Refine the model by introducing nonlinearitiesas desired.

• Important:

- Use reliable and well-understood models.

- Obtain accurate solutions of the models.\"", u ",/

NECESSARY FOR THE INTERPRETATIONOF RESULTS

Thpic One 1-11

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PROBLEM IN NATURE

MODELING

MODEL:We model kinematic conditions

constitutive relationsboundary conditionsloads

SOLVE

INTERPRETATION OFRESULTS

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A TYPICAL NONLINEARPROBLEM

Material: Mild Steel

POSSIBLEQUESTIONS:

Yield Load?

Limit Load?

Plastic Zones?

Residual Stresses?

Yielding whereLoads are Applied?

Creep Response?

Permanent Deflections?

POSSIBLE ANALYSES

Plastic Plasticanalysis analysis

(Small deformations) (Large deformations)

Linear elasticanalysis

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Determine:Total Stiffness;Yield Load

Determine:Sizes and Shapesof Plastic Zones

Determine:Ultimate Load

Capacity

Topic One 1-13

CLASSIFICATION OFNONLINEAR ANALYSES

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1) Materially-Nonlinear-Only (M.N.O.)analysis:

• Displacements are infinitesimal.

• Strains are infinitesimal.

• The stress-strain relationship isnonlinear.

Example:

/J::----.........,.... - P/2

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Material is elasto-plastic..1L

~L < 0.04

• As long as the yield point has notbeen reached, we have a linear analysis.


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2) Large displacements / large rotationsbut small strains:

• Displacements and rotations arelarge.

• Strains are small.

• Stress-strain relations are linearor nonlinear.

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Example:

y

y'

x.1LI·

a'T < 0.04

• As long as the displacements arevery small, we have an M.N.O.analysis.

3) Large displacements, large rotations,large strains:

• Displacements are large.

• Rotations are large.

• Strains are large.

• The stress-strain relation isprobably nonlinear.

Topic One 1-15

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y

Example:

DTransparency

1-16

x

• This is the most general formulationof a problem, considering nononlinearities in the boundaryconditions.


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4) Nonlinearities in boundary conditions

Contact problems:

'1)e--------A~~

--I l-Gap d

• Contact problems can arise with largedisplacements, large rotations,materially nonlinear conditions, ...

Example: Bracket analysis

All dimensions in inches Elasto-plastic materialmodel:

1.5

o

3

thickness = 1 in.

26000psi

Isotropic hardening

e

Finite element model: 36 element mesh

• All elements are a-nodeisoparametric elements

Line of?+--+---+--+--+symmetry

R

Three kinematic formulations are used:

• Materially-nonlinear-only analysis(small displacements/smallrotations and small strains)

• Total Lagrangian formulation(large displacements/largerotations and large strains)

• Updated Lagrangian formulation(large displacements/largerotations and large strains)

Thpic One 1-17

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However, different stress-strain lawsare used with the total and updatedLagrangian formulations. In this case,

• The material law used in conjunction with the total Lagrangianformulation is actually notapplicable to large strain situations(but only to large displ., rotation/small strain conditions).

• The material law used in conjunction with the updated Lagrangianformulation is applicable to largestrain situations.

We present force-deflection curvescomputed using each of the threekinematic formulations and associatedmaterial laws:

15000T.L.

Force(Ibs) u.L.J.

10000M.N.O.

5000

O+----+----+--o 1 2

Total deflection between points ofload application (in)

The deformed mesh corresponding toa load level of 12000 Ibs is shownbelow (the U.L.J. formulation is used).

undeformed A...s-....-r-,,--..,.......,

mesh~ .= = -1

,...-.,...--r---r-«''''' I JI I I I \ "Jr':,I/T"io--.-4.L'.I- _ -+- _ -+- _ -+_'rI I I II---+---+--I I II---+---+-I I I

_~s-deformed mesh

Topic One 1-19

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ComputerAnimationPlate with hole

TIME:· 8LOAD· 8.8 MPA

TIME: • 41LOAD· 512.5 MPA

TIME: • 52LOAD· eS8.S MPA

TIME ,LOAD MPA

t

TIME, 13llLOAD I 325llll MPA

~

/ "/ "

/ "/ "

/ "/ "

/ "

" /" /" /" /" /" /" /v

TIME 3llllLOAD 751100 MPA

~

/ "/ "

/ "/ "

/ "/ "

/ "/ "/. ~

~ ~

" /" /" /" /" /" /" /" /v

Topic One 1-21

ComputerAnimationDiamond shapedframe


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THE BASIC APPROACH OF ANINCREMENTAL SOLUTION

• We consider a body (a structure orsolid) subjected to force anddisplacement boundary conditions thatare changing.

• We describe the externally appliedforces and the displacement boundaryconditions as functions of time.

time time

Since we anticipate nonlinearities,we use an incremental approach,measured in load steps or time steps

Topic One 1-23

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time

When the applied forces anddisplacements vary

- slowly, meaning that the frequenciesof the loads are much smaller thanthe natural frequencies of thestructure, we have a static analysis;

- fast, meaning that the frequenciesof the loads are in the range of thenatural frequencies of the structure,we have a dynamic analysis.

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Meaning of time variable

• Time is a pseudo-variable, onlydenoting the load levelinNonlinear static analysis with timeindependent material properties

Run 1

at = 2.0

2.0 4.0 time

1.02.0

at = 1.0~-I-----+-----

time

R200.014----4100.0

Example:Transparency

1-29

Identically the sameresults are obtained inRun 1 and Run 2

Time is an actual variable

- in dynamic analysis

- in nonlinear static analysis withtime-dependent material properties(creep)

Now dt must be chosen carefully withrespect to the physics of the problem,the numerical technique used and thecosts involved.

At the end of each load (or time)step, we need to satisfy the threebasic requirements of mechanics:

• Equilibrium

• Compatibility

• The stress-strain law

This is achieved - in an approximatemanner using finite elements-by theapplication of the principle of virtualwork.

Topic One 1-25

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1·26 Introduction to Nonlinear Analysis

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We idealize the body as anassemblage of finite elements andapply the principle of virtual work to theunknown state at time t+.!1t.

H.1tR =

~vector ofexternally appliednodal point forces(these include theinertia forces indynamic analysis)

H.1tF

~vector ofnodal point forcesequivalent to theinternal elementstresses

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• Now assume that the solution at timet is known. Hence ~iJ-t tv, ... areknown.

• We want to obtain the solutioncorresponding to time t+.!1t (Le., forthe loads applied at time t+.!1t).

• For this purpose, we solve in staticanalysis

tK .!1U = H.1tR - tF- - -H.1tU . tu + .!1U

More generally, we solve

using

Topic One 1-27

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Slide1-1

Slide1-2

R ~ 100in.

h ~ 1 in.

E =1.0x10' Ib/in2

II =113

O'y~4.1 xlO'lb/in2

E1 =2.0x10slb/in2

f = 9.8 xlO-2 lb/in3

Initial imperfection : Wj (¢) =ShPll C05¢

Analysis of spherical shell under uniformpressure loading p

I

I

I

I

LTwenty 8-node aXlsymmetnc els.

p deformation dependent

Finite element model

Thpic One 1-29

'EI~Slic 'Plasl,c T.l.(E'P, III

,EI~slic·PI~slic small d,sp. IE-PI

•.:,( .. 0.9 8'P2V'

••••• ll:

....::.--- EI~sl;c sm~1I disp. (E 1 _

Slide1-3

0·80·60·40·2

EI~slic T.l(E, Tl )

o

'·0

0·2

06

0.4

0·8

PRES~

RADIAL DlSPlAC~NT AT ~.O - ,ncheS

Static response ofperfect (5 =0) shell

0·80·60·40·2o

0·8

/E'P Slide

0·61-4

\ E. T.l.

0·4PRES~

PIPer

0·2

RADIAL DlSPlACEf'lENT AT ~·O - ,ncheS

Static response of imperfect (5 =0.1) shell


Slide1-5

PRESSlRE

PIPer

o 0·2 0·4 0·6

RADIAL DISPLACEMENT AT ~ =0inches

Elastic-plastic static buckling behavior of theshell with various levels of initial imperfection

Slide1-6 8

MEAN

DISPLACEtvENT "

6

4

2

o

:-:i 'q p

0.0'

0."

" !

2 4 6 8TIME T

Dynamic response of perfect (~ = 0)shell under step external pressure.

10

TIME T

Dynamic response of imperfect (15 = 0.1)shell under step external pressure.

Slide1-7

Topic One 1-31

10x16'Slide

5" 0.1 1·88

6IoIEAN

DISPLACEr-£NT ..

4

2

0 2 4 6 8 10TIME T

Elastic-plastic dynamic response of imperfect (0 =0. J) shell


Slide1-9

0·8

.. Static unstable

• Dynamic unstable

o Dynamic stable

0·4

0·2

o 0.1 0·2 0·3 0·4

A~L1TUDE OF IMPERFECTION '6

Effect of initial imperfections on the elastic-plasticbuckling load of the shell

MIT OpenCourseWare http://ocw.mit.edu

Resource: Finite Element Procedures for Solids and Structures Klaus-Jürgen Bathe

The following may not correspond to a particular course on MIT OpenCourseWare, but has been provided by the author as an individual learning resource.

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